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Homework 6

This is the task corresponding to homework 6.


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Definitions File

theory Defs
  imports "HOL-IMP.BExp" "HOL-IMP.Star"

  com = SKIP
      | Assign vname aexp       ("_ ::= _" [1000, 61] 61)
      | Seq    com  com         ("_;;/ _"  [60, 61] 60)
      | If     bexp com com     ("(IF _/ THEN _/ ELSE _)"  [0, 0, 61] 61)
(* begin mod *)
      | Loop  com bexp
(* end mod *)

  big_step :: "com \<times> state \<Rightarrow> state \<Rightarrow> bool" (infix "\<Rightarrow>" 55)
Skip: "(SKIP,s) \<Rightarrow> s" |
Assign: "(x ::= a,s) \<Rightarrow> s(x := aval a s)" |
Seq: "\<lbrakk> (c\<^sub>1,s\<^sub>1) \<Rightarrow> s\<^sub>2;  (c\<^sub>2,s\<^sub>2) \<Rightarrow> s\<^sub>3 \<rbrakk> \<Longrightarrow> (c\<^sub>1;;c\<^sub>2, s\<^sub>1) \<Rightarrow> s\<^sub>3" |
IfTrue: "\<lbrakk> bval b s;  (c\<^sub>1,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t" |
IfFalse: "\<lbrakk> \<not>bval b s;  (c\<^sub>2,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t" |
(* begin mod *)
LoopFalse: "\<lbrakk> (c,s) \<Rightarrow> t; \<not>bval b t \<rbrakk> \<Longrightarrow> (Loop c b,s) \<Rightarrow> t" |
LoopTrue: "\<lbrakk> (c,s\<^sub>1) \<Rightarrow> s\<^sub>2;  bval b s\<^sub>2; (Loop c b, s\<^sub>2) \<Rightarrow> s\<^sub>3 \<rbrakk>
  \<Longrightarrow> (Loop c b, s\<^sub>1) \<Rightarrow> s\<^sub>3"
(* end mod *)

text\<open> Proof automation: \<close>

declare big_step.intros [intro]

lemmas big_step_induct = big_step.induct[split_format(complete)]

subsection "Rule inversion"

inductive_cases SkipE[elim!]: "(SKIP,s) \<Rightarrow> t"
inductive_cases AssignE[elim!]: "(x ::= a,s) \<Rightarrow> t"
inductive_cases SeqE[elim!]: "(c1;;c2,s1) \<Rightarrow> s3"
inductive_cases IfE[elim!]: "(IF b THEN c1 ELSE c2,s) \<Rightarrow> t"
inductive_cases WhileE[elim]: "(Loop c b,s) \<Rightarrow> t"

subsection "List setup"

text \<open>
  In the following, we use the length of lists as integers
  instead of natural numbers. Instead of converting @{typ nat}
  to @{typ int} explicitly, we tell Isabelle to coerce @{typ nat}
  automatically when necessary.
declare [[coercion_enabled]]
declare [[coercion "int :: nat \<Rightarrow> int"]]

text \<open>
  Similarly, we will want to access the ith element of a list,
  where @{term i} is an @{typ int}.
fun inth :: "'a list \<Rightarrow> int \<Rightarrow> 'a" (infixl "!!" 100) where
"(x # xs) !! i = (if i = 0 then x else xs !! (i - 1))"

text \<open>
  The only additional lemma we need about this function
  is indexing over append:
lemma inth_append [simp]:
  "0 \<le> i \<Longrightarrow>
  (xs @ ys) !! i = (if i < size xs then xs !! i else ys !! (i - size xs))"
by (induction xs arbitrary: i) (auto simp: algebra_simps)

text\<open> We hide coercion @{const int} applied to @{const length}: \<close>

abbreviation (output)
  "isize xs == int (length xs)"

notation isize ("size")

subsection "Instructions and Stack Machine"

datatype instr =
  LOADI int | LOAD vname | ADD | STORE vname |
  JMP int | JMPLESS int | JMPGE int

type_synonym stack = "val list"
type_synonym config = "int \<times> state \<times> stack"

abbreviation "hd2 xs == hd(tl xs)"
abbreviation "tl2 xs == tl(tl xs)"

fun iexec :: "instr \<Rightarrow> config \<Rightarrow> config" where
"iexec instr (i,s,stk) = (case instr of
  LOADI n \<Rightarrow> (i+1,s, n#stk) |
  LOAD x \<Rightarrow> (i+1,s, s x # stk) |
  ADD \<Rightarrow> (i+1,s, (hd2 stk + hd stk) # tl2 stk) |
  STORE x \<Rightarrow> (i+1,s(x := hd stk),tl stk) |
  JMP n \<Rightarrow>  (i+1+n,s,stk) |
  JMPLESS n \<Rightarrow> (if hd2 stk < hd stk then i+1+n else i+1,s,tl2 stk) |
  JMPGE n \<Rightarrow> (if hd2 stk >= hd stk then i+1+n else i+1,s,tl2 stk))"

  exec1 :: "instr list \<Rightarrow> config \<Rightarrow> config \<Rightarrow> bool"
     ("(_/ \<turnstile> (_ \<rightarrow>/ _))" [59,0,59] 60)
  "P \<turnstile> c \<rightarrow> c' =
  (\<exists>i s stk. c = (i,s,stk) \<and> c' = iexec(P!!i) (i,s,stk) \<and> 0 \<le> i \<and> i < size P)"

lemma exec1I [intro, code_pred_intro]:
  "c' = iexec (P!!i) (i,s,stk) \<Longrightarrow> 0 \<le> i \<Longrightarrow> i < size P
  \<Longrightarrow> P \<turnstile> (i,s,stk) \<rightarrow> c'"
by (simp add: exec1_def)

  exec :: "instr list \<Rightarrow> config \<Rightarrow> config \<Rightarrow> bool" ("(_/ \<turnstile> (_ \<rightarrow>*/ _))" 50)
  "exec P \<equiv> star (exec1 P)"

declare star.step[intro]

lemmas exec_induct = star.induct [of "exec1 P", split_format(complete)]

code_pred exec1 by (metis exec1_def)

  "{(i,map t [''x'',''y''],stk) | i t stk.
    [LOAD ''y'', STORE ''x''] \<turnstile>
    (0, <''x'' := 3, ''y'' := 4>, []) \<rightarrow>* (i,t,stk)}"

subsection\<open> Verification infrastructure \<close>

text\<open> Below we need to argue about the execution of code that is embedded in
larger programs. For this purpose we show that execution is preserved by
appending code to the left or right of a program. \<close>

lemma iexec_shift [simp]:
  "((n+i',s',stk') = iexec x (n+i,s,stk)) = ((i',s',stk') = iexec x (i,s,stk))"
by(auto split:instr.split)

lemma exec1_appendR: "P \<turnstile> c \<rightarrow> c' \<Longrightarrow> P@P' \<turnstile> c \<rightarrow> c'"
by (auto simp: exec1_def)

lemma exec_appendR: "P \<turnstile> c \<rightarrow>* c' \<Longrightarrow> P@P' \<turnstile> c \<rightarrow>* c'"
by (induction rule: star.induct) (fastforce intro: exec1_appendR)+

lemma exec1_appendL:
  fixes i i' :: int
  "P \<turnstile> (i,s,stk) \<rightarrow> (i',s',stk') \<Longrightarrow>
   P' @ P \<turnstile> (size(P')+i,s,stk) \<rightarrow> (size(P')+i',s',stk')"
  unfolding exec1_def
  by (auto simp del: iexec.simps)

lemma exec_appendL:
  fixes i i' :: int
 "P \<turnstile> (i,s,stk) \<rightarrow>* (i',s',stk')  \<Longrightarrow>
  P' @ P \<turnstile> (size(P')+i,s,stk) \<rightarrow>* (size(P')+i',s',stk')"
  by (induction rule: exec_induct) (blast intro!: exec1_appendL)+

text\<open> Now we specialise the above lemmas to enable automatic proofs of
@{prop "P \<turnstile> c \<rightarrow>* c'"} where @{text P} is a mixture of concrete instructions and
pieces of code that we already know how they execute (by induction), combined
by @{text "@"} and @{text "#"}. Backward jumps are not supported.
The details should be skipped on a first reading.

If we have just executed the first instruction of the program, drop it: \<close>

lemma exec_Cons_1 [intro]:
  "P \<turnstile> (0,s,stk) \<rightarrow>* (j,t,stk') \<Longrightarrow>
  instr#P \<turnstile> (1,s,stk) \<rightarrow>* (1+j,t,stk')"
by (drule exec_appendL[where P'="[instr]"]) simp

lemma exec_appendL_if[intro]:
  fixes i i' j :: int
  "size P' <= i
   \<Longrightarrow> P \<turnstile> (i - size P',s,stk) \<rightarrow>* (j,s',stk')
   \<Longrightarrow> i' = size P' + j
   \<Longrightarrow> P' @ P \<turnstile> (i,s,stk) \<rightarrow>* (i',s',stk')"
by (drule exec_appendL[where P'=P']) simp

text\<open> Split the execution of a compound program up into the excution of its
parts: \<close>

lemma exec_append_trans[intro]:
  fixes i' i'' j'' :: int
"P \<turnstile> (0,s,stk) \<rightarrow>* (i',s',stk') \<Longrightarrow>
 size P \<le> i' \<Longrightarrow>
 P' \<turnstile>  (i' - size P,s',stk') \<rightarrow>* (i'',s'',stk'') \<Longrightarrow>
 j'' = size P + i''
 P @ P' \<turnstile> (0,s,stk) \<rightarrow>* (j'',s'',stk'')"
by(metis star_trans[OF exec_appendR exec_appendL_if])

declare Let_def[simp]

subsection "Compilation"

fun acomp :: "aexp \<Rightarrow> instr list" where
"acomp (N n) = [LOADI n]" |
"acomp (V x) = [LOAD x]" |
"acomp (Plus a1 a2) = acomp a1 @ acomp a2 @ [ADD]"

lemma acomp_correct[intro]:
  "acomp a \<turnstile> (0,s,stk) \<rightarrow>* (size(acomp a),s,aval a s#stk)"
by (induction a arbitrary: stk) fastforce+

fun bcomp :: "bexp \<Rightarrow> bool \<Rightarrow> int \<Rightarrow> instr list" where
"bcomp (Bc v) f n = (if v=f then [JMP n] else [])" |
"bcomp (Not b) f n = bcomp b (\<not>f) n" |
"bcomp (And b1 b2) f n =
 (let cb2 = bcomp b2 f n;
        m = (if f then size cb2 else (size cb2::int)+n);
      cb1 = bcomp b1 False m
  in cb1 @ cb2)" |
"bcomp (Less a1 a2) f n =
 acomp a1 @ acomp a2 @ (if f then [JMPLESS n] else [JMPGE n])"

  "bcomp (And (Less (V ''x'') (V ''y'')) (Not(Less (V ''u'') (V ''v''))))
     False 3"

lemma bcomp_correct[intro]:
  fixes n :: int
  "0 \<le> n \<Longrightarrow>
  bcomp b f n \<turnstile>
 (0,s,stk)  \<rightarrow>*  (size(bcomp b f n) + (if f = bval b s then n else 0),s,stk)"
proof(induction b arbitrary: f n)
  case Not
  from Not(1)[where f="~f"] Not(2) show ?case by fastforce
  case (And b1 b2)
  from And(1)[of "if f then size(bcomp b2 f n) else size(bcomp b2 f n) + n"
       And(2)[of n f] And(3)
  show ?case by fastforce
qed fastforce+

consts ccomp :: "com \<Rightarrow> instr list"


Template File

theory Submission
  imports Defs

fun ccomp :: "com \<Rightarrow> instr list" where
"ccomp SKIP = []" |
"ccomp (x ::= a) = acomp a @ [STORE x]" |
"ccomp (c\<^sub>1;;c\<^sub>2) = ccomp c\<^sub>1 @ ccomp c\<^sub>2" |
"ccomp (IF b THEN c\<^sub>1 ELSE c\<^sub>2) =
  (let cc\<^sub>1 = ccomp c\<^sub>1; cc\<^sub>2 = ccomp c\<^sub>2; cb = bcomp b False (size cc\<^sub>1 + 1)
   in cb @ cc\<^sub>1 @ JMP (size cc\<^sub>2) # cc\<^sub>2)"
| "ccomp _ = undefined"

value "ccomp (Loop (''u'' ::= Plus (V ''u'') (N 1)) (Less (N 0) (V ''u'')))"

lemma ccomp_bigstep:
  "(c,s) \<Rightarrow> t \<Longrightarrow> ccomp c \<turnstile> (0,s,stk) \<rightarrow>* (size(ccomp c),t,stk)"


Check File

theory Check
  imports Submission

lemma ccomp_bigstep: "(c,s) \<Rightarrow> t \<Longrightarrow> ccomp c \<turnstile> (0,s,stk) \<rightarrow>* (size(ccomp c),t,stk)"
  by (rule Submission.ccomp_bigstep)


Terms and Conditions